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In a locality, two-thirds of the people have cable TV, one- fifth have VCR, and one-tenth have both. What is the fraction of people having either cable -TV or VCR?

1. $19/30$
2. $2/3$
3. $17/30$
4. $23/30$

Given that,

• $n(\text{Cable TV)}=\frac{2}{3}$
• $n(\text{VCR})=\frac{1}{5}$
• $n(\text{Cable TV$\cap$VCR})=\frac{1}{10}$

Lets draw the venn diagram.

Now, the fraction of people having either Cable TV or VCR

$= n(\text{Cable TV$\cup$VCR})-n(\text{Cable TV$\cap$VCR})$

$= n(\text{Cable TV)+n(VCR)}-n(\text{Cable TV$\cap$VCR})-n(\text{Cable TV$\cap$VCR})$

$=\frac{2}{3}+\frac{1}{5}-\frac{1}{10}-\frac{1}{10}$

$=\frac{2}{3}+\frac{1}{5}-\frac{2}{10}$

$=\frac{2}{3}+\frac{1}{5}-\frac{1}{5}$

$=\frac{2}{3}$

(OR)

The fraction of people having either Cable TV or VCR

$= n(\text{Cable TV})-n(\text{Cable TV$\cap$VCR})+n(\text{VCR})-n(\text{Cable TV$\cap$VCR})$

$=\frac{2}{3}-\frac{1}{10}+\frac{1}{5}-\frac{1}{10}$

$=\frac{2}{3}$

Correct Answer : $\text{B}$

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